The armature does not move with the mesh
your geometry is unsuitable for rigging. Imagine, that bones moving with vertexes. In your case is impossible bending mesh by bones correctly, because mesh have really bad topology for animation. And this is the reason why Blender can't assign vertexes to bones and give error message.

Normally you can use "Automatic weights" feature in Blender. Select mesh, press "SHIFT+LMB" select armature. After press "CTRL+P" assign mesh to bones. But this doesn't work in your case.



This is how mesh must look for use in animation:
..as you can see, it work


You can control, how armature move with vertex by "Weight paint". Every bone have vertex group with the same name. Here you can set how strong are vertexes controlled by which bone.


I hope this help you. Also you can find a dozens tutorials about this theme all over the internet.